If you are doing surface integrals starting from single dimensional infinitesimals, then you need two integrals because areas are two dimensional. If your integral is formed such that it already happens to start with an infinitesimally thick slice of the surface then you don't need two integrals because you can integrate that slice along a linear dimension to form the surface. Essentially, with two integrals you are building up from linear dimensions, to a slice, and to an area.

If you are doing surface integrals starting from single dimensional infinitesimals, then you need two integrals because areas are two dimensional. If your integral is formed such that it already happens to start with an infinitesimally thick slice of the surface then you don't need two integrals because part of the work is already finished. 

Essentially, with two integrals you are building up from linear dimensions, to a slice, and to an area. The first integral integrates any one a slice of the surface. The second integral integrates all the slices to form the entire surface.

It progresses the same way to volume integrals, where the surface you have after two integrals is actually a cross section of the volume. Then the third integral adds up all the cross sections to get you the volume.